The Talmud (Megillah 7a) notes that one of the proofs that Megillas Esther was written with ruach hakodesh (see Introduction) is that no human writer could possibly know that the Jews did not take any spoils.
Rashi writes that the Jews had rights to the spoils, but decided to wave those rights, and give the spoils to the king in order to maintain friendly relations with the palace.
The Dena Pishra writes that they did not take spoils because they did not want others to think that the Jews’ motivation was financial.
In M’aarchei Lev, Rav Moshe Schwab writes that since this was the property of Amalek, it was forbidden to take, as was the case for Shaul (Shmuel 1 15:3). and this is why the Jews refrained from doing so here.
In fact, the Binyan Ariel and Nachal Eshkol write that the Jews’ self-control in this incident was a tikun for the sin of Shaul in sparing (Shmuel 1 15:9) Amalek’s property.
Interestingly, the M’lo HaOmer and Me’am Loez both note that the initial letters of the ninth, tenth, eleventh and twelfth words of this verse, uvabeeza lo shalchu es (“and from their spoils they did not send”) can be rearranged to spell Shaul.
The Sfas Emes writes that the Jews took the spoils, but destroyed them in an effort to not benefit from the property.
However, R’ Yitzchak Yeruchem Diskin writes in Ohelim that Jews have an obligation to take the property of Amalek and destroy it, but did not do so here. The reason was that the Talmud (Megillah 16a) considers Haman to have been a slave. As such, he relinquished all rights to personal belongings. This includes his children. This also answers the question of how his grandchildren could study Torah in Bnei Brak if Amalek is never allowed to join the Jewish people. Such is not the case for his grandchildren because of his status of being a slave.
Megillas Seris adds another reason they did not take the spoils – they only had one day to kill Amalek, and they did not want to run the risk of missing the opportunity to fulfill this mitzva. In the course of performing a mitzva, they totally ignored anything ancillary to killing out their enemies.
The Gerrer Rebbe notes that matanos la’evyonim, the Halachic (Shulchan Aruch Orach Chaim 695:4) injunction to donate to the poor on Purim is in honor of the impoverished Jews of the time not taking the spoils of their enemies, despite their needs.